MATHEMATICAL ABSTRACTION OF PRE-SERVICE MATHEMATICS TEACHERS IN LEARNING NON-CONVENTIONAL MATHEMATICS CONCEPTS A Dissertation Submitted in Partial Fulfillment of the Requirements for Doctor of Mathematics Education
MATHEMATICAL ABSTRACTION OF PRE-SERVICE MATHEMATICS TEACHERS
IN LEARNING NON-CONVENTIONAL MATHEMATICS CONCEPTS
A Dissertation Submitted in Partial Fulfillment
of the Requirements for Doctor of Mathematics Education
Farida Nurhasanah NIM: 1201426 STUDY PROGRAM OF MATHEMATICS EDUCATION SCHOOL OF POSTGRADUATE STUDIES UNIVERSITAS PENDIDIKAN INDONESIA 2018
ACKNOWLEDGMENTS
I would like to thank Allah SWT for giving me the best parents in the world who have an everlasting love for their daughter. Without their affection I could not reach this stage, completing my dissertation. I have never imagined that I could finish my thesis by having three periods of pregnancies and taking care of two babies and two children, but Allah SWT gives the strength through my parents’ assistance. I will walk in a humble manner upon receiving my degree, knowing that it was Allah SWT who enabled me to reach this milestone in my life.
I owe my deepest gratitude to my supervisor, Professor Yaya S. Kusumah, MSc., Ph.D., the best supervisor a doctorate mathematics student can have. You have taught me by your own expert model how to create not only good but the best writing article through the comments and revision in the draft. Through a long discussion in your comfortable room, you show me how to be a patient, humble, and humorous scientist but still have a good critical thinking. You have a special skill of reviewing articles and selecting the words to express ideas in wonderful sentences. Most importantly, you bring me to experience the academic atmosphere in international forum that inspired me a lot. I am glad that I have had the opportunity to be one of those privileged to work with you.
I extend my sincere appreciation to Prof. Didi Suryadi, M.Ed., who has helped me in answering difficult questions that I ask to myself. My sincere appreciation also goes to Prof. Turmudi, M.Sc., M.Ed., Ph.D., Dr. Sufyani Prabawanto, M.Ed., Dr. Elah Nurlaelah, M.Si., and Eyus Sudihartinih, M.Pd., for all their valuable help, support, and unwavering assistance throughout the completion of this study.
I would like also to thank my extended family, my little sister, the big family of Mbah Hadiwijaya, Mbah H. Mawardi, and the big family of Mbah H. Madmasum, the big family of Marimun and Rasinah for their encouragement and prayer. Most of all, I would like to thank my loving husband, you have given me so much and loved me so well. I would not have accomplished this without you beside me.
Finally, I cannot thank Prof. Jozua Sabandar enough for encouraging me to pursue something that I never imagine before in academic level. He always helps me when I asked for it. I am impressed by the everlasting, enthusiastic attitude that he has for guiding and motivating students. I would not have reached this point in my career without the influence of Prof. Jozua Sabandar, M.A., Ph.D. He not only served as my advisor during my Master and Doctorate studies at Universitas Pendidikan Indonesia, but he has become a lifelong best friend.
The last but not least, I want to extend my high gratitude to my colleagues in Mathematics Education Department, Sebelas Maret University, for their sincere giving me the opportunity to take my doctorate degree. Also to my fellow friends, Nurfadillah Siregar for every single support that you gave, for Ibu Maria, Puji Lestari, Pak Joko, Pak Cecep, Ibu Laila, Pak Mujib, and all friends that cannot be mentioned one by one in this page, thanks for the friendship. Thanks also to all the committee of SEMDIKMAT 2016, thank you for trusting me! Thanks to Mr. Juanda for the opportunity that you gave to me during my refreshing time in finishing my dissertation.
Bandung, January 2018
Farida Nurhasanah
DEDICATION
I dedicate my dissertation to my beloved parents, Hartini and Kasan for their unlimited loves. This dissertation is a proof of a woman’s love for her husband, Agus Fitriandi, and the
children Mote, Moza, Mota, and baby Moze.
I dedicated this dissertation to every special person in my life who has been creating mosaics
of my wings so that I can fly high out of the sky.
I dedicate this dissertation to myself who deserved to be happy for the choices of life that have
been taken.
This love has guided me to find a long journey of a wonderful time of study in a wonderful
place with wonderful people.
Nurhasanah, F. (2018). Mathematical Abstraction of Pre-Service Mathematics Teachers in Learning Non-Conventional Mathematics Concepts.
ABSTRACT
One of the important processes in learning and teaching mathematics is mathematical abstraction. Having rich experiences with this will be beneficial for pre-service mathematics teachers in understanding students’ learning processes as well as for designing mathematics instruction. Mathematical abstraction is a cognitive process that takes place in learners’ mind. So, to be able to analyze the process, a learning context needs to be designed specifically for triggering this process. A theoretical framework that also provides methodological tools for investigating this process is Abstraction in Context (AiC). Using AiC framework, this study aims to investigate three particular issues: (1) the abstraction processes of pre-service mathematics teachers when they learn non-conventional mathematics concepts; (2) mathematical abstraction levels raised by pre-service mathematics teachers in learning non- conventional mathematics concepts; and (3) relationship between the abstraction process of pre-service mathematics teachers in learning non-conventional mathematics concepts and their performance in learning conventional mathematics concepts. This is a qualitative study that rely on the aims of the study. This study employed a grounded study using AiC framework and RBC model for answering the first and the second issue. In addition, a case study designed using descriptive statistics is used for answering the last issue. A context comprised a lesson plan, learning activities, a set of module, and classroom setting, while learning environment were designed as part of AiC framework. The topic of Parallel Coordinates was selected especially for the design. The study was conducted in six months, involving 45 participants in Analytic Geometry course. All participants have to construct six knowledge elements in Parallel Coordinates topic after learning concept of Cartesian coordinate, then taking a prior knowledge test on Cartesian coordinate concept. The data were collected through video recording, students’ worksheet, test, and field notes. Related to the first issue, the abstraction process of pre-service mathematics teachers in learning low-level knowledge elements mostly takes place in group context. Whereas the abstraction process of them in learning high-level knowledge elements takes place in classroom context. In both contexts researcher find that reducing abstraction can help participants in constructing new mathematical knowledge. Regarding the second issue, Pre-service mathematics teachers as the participants in this study are can be categorized into three level of abstraction, less than half of participants are in level 1 (perceptual abstraction) and in level 2 (internalization). The rest of participants are in level 4 (second level of interiorization). There are no participants belong to level 3 (interiorization) while the others are still in pre-level of perceptual abstraction and in transition to level 1 and level 3. With regard to the final issue, there is a positive linear association between participants’ scores in Parallel Coordinates test and participants’ scores in Analytic Geometry test.
Keywords: Mathematical Abstraction, Parallel Coordinates, Non-Conventional Mathematics Concept, Abstraction Level.
Nurhasanah, F. (2017). Mathematical Abstraction of Pre-Service Mathematics Teachers in Learning Non-Conventional Mathematics Concept.
ABSTRAK
Abstraksi merupakan salah satu proses yang penting dalam pembelajaran matematika. Guru yang kaya pengalaman dengan proses abstraksi akan lebih mudah memahami proses belajar siswa sehingga dapat merancang proses pembelajaran yang sesuai dengan karakteristik siswanya. Abstraksi merupakan proses kognitif yang berlangsung di benak seorang individu yang mempelajari konsep baru, sehingga proses tersebut hanya dapat dianalisis ketika pembelajaran dirancang khusus untuk memicu munculnya proses tersebut. Terkait dengan hal tersebut penelitian tentang abstraksi membutuhkan kerangka teoretis dan metodologi khusus. Salah satu kerangka teoretis yang dapat digunakan untuk mengkaji proses abstraksi adalah Abstraction in Context (AiC). Kerangka teoretis ini dilengkapi model RBC (Recognizing, Building -with, Construction) untuk menganalisis data kualitatif yang diperoleh ketika proses abstraksi berlangsung. Dengan menggunakan kerangka AiC, penelitian ini bertujuan untuk mengkaji tiga isu: (1) proses abstraksi calon guru matematika ketika mereka mempelajari konsep matematika non-konvensional; (2) level abstraksi matematis calon guru matematika dalam belajar konsep matematika non-konvensional; dan (3) hubungan antara proses abstraksi calon guru matematika dalam belajar konsep matematika non-konvensional dan hasil belajar mahasiswa calon guru matematika dalam belajar konsep matematika konvensional. Penelitian ini merupakan penelitian kualitatif yang menggunakan desain grounded study dengan menggunakan kerangka AiC dan model RBC untuk menjawab pertanyaan pertama dan kedua. Sedangkan desain studi kasus menggunakan statistik deskriptif digunakan untuk menjawab pertanyaan terakhir. Berdasarkan kerangka AiC, rencana pembelajaran, lembar aktivitas mahasiswa, modul, setting kelas, dan lingkungan belajar dirancang dengan memlilih topik Koordinat Parallel sebagai konsep yang harus dikonstruksi. Penelitian dilakukan selama enam bulan, melibatkan 45 peserta pada kelas Geometri Analitik. Semua peserta harus membangun enam elemen pengetahuan dalam topik Koordinat Paralel. Data dikumpulkan melalui rekaman video, lembar kerja siswa, tes, wawancara, dan catatan lapangan. Proses abstraksi calon guru matematika sebagian besar berlangsung pada konteks kelompok saat membangun elemen pengetahuan yang memiliki tingkat abstraksi yang rendah dan terjadi pada konteks kelas ketika elemen pengetahuan memiliki tingkat abstraksi yang tinggi. Dalam penelitian ini, isu reduksi abstraksi ditemukan berperan penting dalam proses mengkonstruksi konsep matematika yang baru. Mengenai isu kedua, calon guru matematika sebagai dalam penelitian ini dapat dikategorikan menjadi tiga level. Sebagian besar peserta berada di level 1 (perceptual abstraction) dan level 2 (internalization). Sisanya berada di level 4 (second level of interiorization ). Tidak ditemukan peserta yang berada di level 3. Masih ditemukan beberapa peserta yang berada pada level pre- perceptual abstraction dan masih dalam transisi ke level 1 dan level 3. Berkaitan dengan isu terakhir, diperoleh korelasi yang positif antara skor pada tes abstraksi matematis dan skor pada tes Geometri Analtik.
Keywords: Abstraksi Matematis, Koordinat Paralel, Konsep Matematika non-konvensional, Level Abstraksi.
A. Rational and Background
The quality of mathematics education is influenced by many factors such as curriculum, facilities, learning environment, and teachers. All these factors are considered as integral parts of learning process in a classroom. In classroom teaching process, teacher becomes the key person who is responsible for students’ learning process (Turnuklu & Yesildere, 2007; Sabandar & Nurhasanah, 2014). It can be stated that in order to have a good quality of mathematics education we need qualified mathematics teachers. To obtain qualified mathematics teachers, professional development programs need to be addressed not only for in-service teachers but also for pre-service teachers.
Mathematical thinking experiences are very important for pre-service mathematics teachers, as these experiences will contribute to the way they teach mathematics. They can use their experiences in mathematical thinking to predict in which part students will face problem and how they can cope with that problem. Mathematical thinking consists of various strands such as reasoning, proofing, representation, communication, problem-solving, abstraction, and reflection. All those strands are essential for pre-service mathematics teachers. Some studies related to mathematical thinking in pre-service mathematics teachers are focused on the issues concerning mathematical problem-solving, representation, communication, creative and critical thinking. However, there are still limited studies on abstraction of pre-service mathematics teachers whereas this thinking process belongs to advanced mathematical thinking (Tall, 2002), as well as fundamental process in mathematics and mathematics learning (Ferari, 2003).
Based on studies by experts such as Schwartz, Dreyfus, & Herskowitz (2009); Kidron, 2008; Ozmantar and Monaghan, (2006); Mitchelmore and White (2004); Hazzan (2003), mathematical abstraction could be classified into empirical abstraction and theoretical abstraction referred to types of mathematics concepts in mathematics learning. Empirical mathematical abstraction processes take place when someone learns fundamental mathematics concepts that mostly part of elementary concepts such as numbers and their operation (Mitchelmore and White, 2004). In more advanced mathematics learning, many mathematics concepts have no counterparts in daily life experiences, for example the square root of negative numbers. Those mathematics concepts are easily founded in more advanced mathematics courses such as Real Analysis or Abstract Algebra. The process of abstraction in learning these types of mathematics concepts involves a process of similarity recognition followed by formalization; it is widely known as theoretical abstraction (Mitchelmore and White, 2004; Nurhasanah, 2010).
Abstraction is a process of mental constructions related to the emergence of new concepts. In order to investigate the emergence of new mathematics concepts for future teachers, a consideration must be made in term of what kind of concept that is suitable to this context. Considering that mathematics schools concepts are part of empirical mathematics concepts, all pre-service mathematics teachers learned all those concepts during their elementary and high school time. Those concepts become no longer “new” for them when they enter university.
For those students who have already mastered the basic (empirical) school mathematics concepts as the objects of abstraction before they come to university, learning the same concepts could be an uninteresting activity. Zaskis (1999) even stated that taking the basic empirical school mathematics concepts as the objects of abstraction could be perceived as boring and even insulting. On the other side, preferring to use advanced mathematical concepts in order to explore their abstraction processes, also considering as a burden for pre- service mathematics teachers, because all those concepts will not be taught to their future students. In addition, because of time constraints, most of curricula for mathematics teachers For those students who have already mastered the basic (empirical) school mathematics concepts as the objects of abstraction before they come to university, learning the same concepts could be an uninteresting activity. Zaskis (1999) even stated that taking the basic empirical school mathematics concepts as the objects of abstraction could be perceived as boring and even insulting. On the other side, preferring to use advanced mathematical concepts in order to explore their abstraction processes, also considering as a burden for pre- service mathematics teachers, because all those concepts will not be taught to their future students. In addition, because of time constraints, most of curricula for mathematics teachers
Ball (1999) provided some evidences from his study that some of pre-service mathematics teachers still have problems in dealing with the basic knowledge of mathematics. Michoux (2013) even stated that many pre-service teachers have no differences with elementary school pupils in terms of their ability in geometry; moreover, they were having similar difficulties and misconception as compared to their pupils. This condition will cause pre-service teachers encounter problem when they become a teacher and meet smart students who ask questions such as “Why 25 + 5 is equal to 30 not equal to 255?” or “Why we have to put zero point at the intersection of x-axis and y-axis?”. Without a profound mathematical background, this type of questions will end up with answer “we just take it for granted”.
In order to build a profound mathematical background for pre-service mathematics teachers, a list of mathematics topics is compulsory for them. Even though probably they will have certain preferences but they have to master all compulsory topics. Unfortunately, intensive study about why some mathematics topics are more important than the others for pre-service mathematics teachers are still limited and debatable (Zaskis, 1999).
There are at least three types of mathematics concepts that are important for pre-service mathematics teachers: (1) elementary mathematics; (2) intermediate mathematics; and (3) advanced mathematics (Suryadi, 2016). Elementary mathematics means all mathematics concepts that are learned by students from elementary until high school level. Intermediate mathematics concepts means part of mathematics concept for undergraduate level such as Calculus, Algebra, Linear Algebra, Discrete Mathematics, while advanced mathematics concepts means mathematics concepts that are part of mathematics topics for graduate students or the newest theory from mathematics researches in various journals or monographs. All those types of mathematics concepts can provide complete experiences for pre-service mathematics teachers in mathematical thinking.
Zaskis (1999) has an idea about mathematics concepts that are not part of one of those types stated in the previous paragraph. She proposed a concept of non-conventional mathematics object for pre-service mathematics teachers for achieving deeper understanding of mathematical concepts or constructing richer schemes. Examples of those concepts are fractional number in non-base-10 numeracy, focus-directric coordinate system, Parallel Coordinates, and Boolean operation.
To provide pre-service mathematics teachers with the experience of constructing mathematical concepts, an alternative that can be done is by providing them with learning experiences of mathematics concepts that they have never learned before in order to stimulate the processes of concept construction. Experiences in the processes of constructing new mathematical concepts can sharpen their thinking processes. In addition it also provides them with skills to adapt to new situation.
In order to decide what new mathematics concepts are suitable for abstraction context, some aspects need to be considered. The concepts must not be too rigor for them; it is not part of elementary mathematics concepts but it still can encourage them to use their mathematical thinking; and help them doing abstraction processes. One of the concepts that could be selected is Parallel Coordinates systems as part of non-conventional concepts.
Based on mathematics school curriculum around the world, Cartesian coordinate is one of the compulsory concepts for high school students; almost all high school students are familiar to Cartesian coordinate system. Probably, freshmen in mathematics education department in university also have in mind that Cartesian coordinate system is the only coordinate system that they know before they learning Polar coordinate in Calculus.
Cartesian coordinate system was the greatest invention at the time when for the first time Euclidean geometry could be represented and treated algebraically and numerically.
Invented by Rene Descartes (1596 – 1650), the Cartesian coordinate system led to the development of Analytic Geometry which is very useful for surveyors and navigators.
In Analytic Geometry, geometrical ideas are presented using a coordinate system in the Euclidean plane. Cartesian coordinate uses two axes, namely X –axis and Y-axis, these are two perpendicular number lines of ℝ where each point on the plane has two coordinates. Here ℝ 2 is the set of ordered pairs of real numbers. Cartesian coordinate system is a basic tool of
Analytic Geometry presented in Euclidean plane. Descartes developed it, starting from two- dimensional space in Euclidean plane with 2 ℝ by describing geometry concepts in term of
numbers. Then it was extended into Analytic Geometry in three-dimensional Euclidean space identified as 3 ℝ . Finally it is generalized into Analytic Geometry in n dimensional space ( ℝ ) ,
where the dimension n can be any natural number. Unfortunately when ≥ 4 for ℝ , Euclidean geometry cannot be used anymore to interpret any object in these dimensions in Analytic Geometry.
The concepts and tools developed in Analytic Geometry, particularly on 2 ℝ are very important to be used in generalization and abstraction for multidimensional geometry in 3 ℝ , ℝ , and Linear Algebra. It is also as a fundamental concept for the notion of dimension in
mathematics. Representing objects in more than 3 dimensions in Cartesian coordinate is complicated. Mostly objects on those dimensions are learned without their representation. This condition leads to difficulties in learning higher dimensions. Study did by Skordoulis, Vitsas, Dafermos and Koleza (2008) found that the limitation on Cartesian coordinate system became epistemological and didactical obstacles for pre-service teachers in Greece in learning the concept of dimensions. The coordinate system becomes an epistemological obstacle when students tend to use the approach with coordinates even though they are working outside of the coordinate system. Cartesian coordinate as didactical obstacles in Greece was identified when many students did not notice the case of interdependence of variables in the case of learning curves equation.
This situation was mainly caused by teaching practice of the Greek mathematics teachers who did not sufficiently analyze the interdependence between variables during teaching. In addition, Greek teachers also did not give emphasis on the parametric equation of the circle and the elips, where the dependence of x and y only one variable.
Based on those situations explained earlier, introducing a new coordinate system could
be one of alternatives that can help pre-service teachers to be able to deal with those problems mentioned before. Considering the problems faced by pre-service mathematics teachers in understanding the concept of dimensions, Parallel Coordinates system could be an appropriate concept for helping them to lead their mathematical abstraction process into higher dimensions. This coordinate system can visualize analytic geometry in ℝ (Inselberg & Dimsdale, 1990). Since its first appearance, Parallel Coordinates system has become a well- known visual multidimensional geometry and visualization for exploratory data analysis (Heinrich & Weiskopf, 2012). This non-conventional mathematics concept can help teachers in strengthening and broadening their mathematics knowledge. This concept could also be very useful for analyzing the process of abstraction of pre-service mathematics teachers. This interesting concept, however, has never been introduced to pre-service mathematics teachers.
Information about how pre-service teachers construct new mathematical knowledge could be very useful in order to analyze mathematical thinking process of future teachers. Then, the result could be used to design strategy for developing their pedagogical content knowledge. For example, how pre-service mathematics teachers learn concept of straight lines in Analytic Geometry using vector approach. It can be traced how much the influence of their prior knowledge when learning the concept without vector approach in the concept of Cartesian coordinate.
Study about pre-service mathematics teachers was focused on their achievement or their performance in learning specific concepts in mathematics or focus on what kind of methods or strategies that can be used to enhance their mathematical thinking skills. Unfortunately, studies that have been conducted about their thinking process in learning mathematics are still limited. So that this study will enrich the field of research in mathematics education in term of pre-service mathematics teachers thinking processes. This study will focus on the process of mathematical abstraction that take place when pre-service mathematics teachers learn concept of Parallel Coordinates with the title is “Mathematical Abstraction of Pre-Service Mathematics Teachers in Learning Non-Conventional Mathematics Concepts ”.
B. Research Questions
The problem that stands out most, from the background presented above, on one side is the lack of information about how pre-service mathematics teachers abstraction process in learning new mathematics concepts. On the other side, new mathematical concepts learned by pre-service mathematics teachers mostly belong to advance mathematics concepts. In order to fill this gap, this study will document more carefully the process of mathematical abstraction for pre-service mathematics teachers in learning new concepts: non-conventional concept.
Below are some general questions guiding this study:
1. How do pre-service mathematics teachers’ abstraction processes take place when they learn non-conventional mathematics concepts?
2. What kind of mathematical abstraction levels that could be raised by pre-service mathematics teachers in learning non-conventional mathematics concepts?
3. To what extent the abstraction process of pre-service teachers in learning non-conventional mathematics concepts could indicate their performance in learning conventional mathematics concepts?
C. Aims of the Study
The aims of this study, as conducted in this research are to explore the mathematical abstraction processes of pre-service mathematics teachers in learning non-conventional mathematics concepts in analytic geometry using RBC + C model initiated by Schwarz, et al. (2009), to investigate their levels of abstraction based on the theory of level abstraction proposed by Hazzan (1999), Battista (2007), Nurhasanah, Sabandar, & Kusumah (2013), and Hong & Kim (2015). Finally, the last but not least aim of the study is to investigate the relationship between the abstraction processes of pre-service mathematics teachers in learning non-conventional concept and their understanding in learning concept of school mathematics which has similar structure.
D. Terminology
Since this study focused on the topic of mathematical abstraction in non-conventional mathematics concepts, it is really important to explain the definition of terms that are used in this study such as “mathematical abstraction” and “non-conventional mathematics”. The term ‘abstraction’ is used in two different contexts. First ‘abstraction’ means as a thinking process (also called as ‘mathematical abstraction’). Here, abstraction is defined as an activity of vertically reorganizing previous mathematical construct within mathematics and by mathematical means so as to lead a construct that is new to the learner in learning process. This definition refers to Schwarz, et al. (2009). Second, the term of ‘abstraction’ as ability related to the term of ‘reducing abstraction’ proposed by Hazzan (2003). This term is based on three different interpretations of levels of abstraction discussed in this study: (a) abstraction level as the quality of the relationships between the object of thought and the thinking person; (b) abstraction level as reflection of the process-object duality; and (c) abstraction level as the degree of complexity of the concept of thought.
The term of non-conventional mathematics concepts in this study is adopted from Zazkis (1999). Non-conventional mathematics concept defined as mathematical concepts that are not part of school mathematics and also not part of advanced mathematical concepts. This concept has a similar structure to that of empirical mathematics concepts but not part of mathematics concepts that are learned by students from elementary to high school level. On the opposite, the term of ‘advanced mathematics concept’ refers to object of mathematics which are not part of empirical mathematics concepts or not part of school mathematics, for example Abstract Algebra, Real Analysis and Geometry from an advance point.
Abstraction in Context (AiC) is a theoretical framework for studying students’ processes of constructing abstract mathematical knowledge. This process occurs in a context that includes specific mathematical curricular and social components as well as a particular learning environment. This framework is completed by a model for describing and analyzing the emergence of mathematical constructs that are new to students. This model called as RBC+C- model which is consisting of three observable epistemic actions: Recognizing, Building -with, Construction, and Consolidation. This model serves as the main methodological tool for AiC (Dreyfus, et al., 2015).
E. Benefits of the Study
Firstly, through this study pre-service mathematics teachers will get benefit in terms of mathematical abstraction knowledge, they will have experiences in doing mathematical abstraction, and they will learn new concept of mathematics that they probably never learn before. They could use all those experiences and knowledge in order to develop their content knowledge and pedagogical content knowledge.
Secondly, this study can help lecturers in mathematics education department to develop their teaching material related to mathematical concepts that must be conveyed to their students. In addition, information about the process of mathematical abstraction of pre-service mathematics teachers will be useful for lecturers to determine an appropriate teaching strategy in order to have more effective instructional processes. Knowing more about students’ thinking processes can improve their teaching aimed at supporting students’ own knowledge construction.
Thirdly, through this study, researchers will have new topics to be explored further. Research on mathematical abstraction is relatively new. This study will use two famous theoretical frameworks in topic of mathematical abstraction that have never been used in any study before. This study contributes both for developing mathematical abstraction theory and developing methodology for conducting research in mathematical abstraction topic.
F. Research Method
This study is a qualitative study which comprised of two phase using grounded theory design and case study design. This study adopted Abstraction in Context (AiC) theoretical framework (Dreyfus et al, 2015) for investigating mathematical abstraction process of pre- service mathematics teachers in learning non-conventional mathematics concept. This framework completed by RBC (Recognizing, Building-with, Construction) model and theory of reducing abstraction for analyzing the abstraction process. Indicators for level of abstraction in this study adopted from Dreyfus et al (2001), Gray & Tall (2007), Battista (2007), Nurhasanah, Sabandar, & Kusumah (2013) and Hong and Kim (2015). Related to the third question, a case study design using statistic descriptive is used to analyze relationship between data from participants’ score test on Analytic Geometry and participants’ score test on Parallel Coordinate.
Referred to the AiC design, on a priori analysis, concept of Parallel Coordinates as one of non-conventional mathematics concepts, was selected to be the main topic on this study based on some consideration: (1) the concept must be relatively new for pre-service Referred to the AiC design, on a priori analysis, concept of Parallel Coordinates as one of non-conventional mathematics concepts, was selected to be the main topic on this study based on some consideration: (1) the concept must be relatively new for pre-service
This study was conducted for six months, in a government university in Bandung, involving 45 participants. The pre-service mathematics teachers were grouped consisted of 4 until 5 participants in each group. The researcher intentionally used the result of prior knowledge test on topic of Cartesian coordinate to be the basis for grouping process. Based on the result of prior knowledge test, the participants were grouped into three categories, high, medium, and low in term their performance on the prior knowledge test. Using this result, the participants were grouped into 11 groups consisted of 7 homogenous groups and 4 heterogeneous groups.
The entire participants have never been in touch with concept of Parallel Coordinates. They have never learned other types of coordinates such as Polar coordinate. Cartesian coordinate system both in 2D and 3D are the only coordinate systems that they have been familiar with. Their understanding of prior knowledge concepts such as Cartesian coordinate and Lines was varied.
There were two lecturers who conducted the lecture, a senior lecturer, and a junior one. However, the junior lecturer is the one who was responsible for the teaching while the role of senior lecturer is supervising the learning process of the students. Considering the content curriculum of Analytic Geometry and times constraint, the lecturers and researcher have agreement that topic of Parallel Coordinates would be delivered in additional lecture combined with tutorial class for Analytic Geometry. This tutorial was held once a week. Topic of Parallel coordinates was delivered after the topic of conics in dimension 3. This topic was delivered in three consecutive weeks, completed by a prior knowledge test before the topic was delivered and abstraction test on the topic of Parallel Coordinates after the topic was delivered. Five types of data were collected in this study: video recording of the classroom activity; field notes; video recording of semi-structured interviews; documents of students’ worksheet; and scores of the test.
The pre-service mathematics teachers here were expected to construct the four key concepts that underlie the learning design in this study in four stages: using worksheet 1, 2, 3, and 4, consecutively. Those worksheets were divided into Stage A, Stage B, Stage C, and Stage D. The relationship of the stages and the key concepts that were constructed in each stage are given below:
Table 1. Key Concepts in Each Stage
No Stages Worksheets
Key Concepts
1 A 1 Construction of Parallel Coordinates in 2D (E A 1 )
A representation of a point A(x.y) in Parallel Coordinates
(E A2 )
2 B 2 Representation of a line ℓ≡ = − 2+3 in 2D Parallel
Coordinates(E B )
3 C 3 Generalization of the representation of a line y = mx + b,
≠ 1 in 2D Parallel Coordinates (E C1 )
Representation of a line y = mx + b, =1 (E C2 )
4 D 4 Representation of intersection of two lines (E D1 ) Representation of two parallel lines (E D2 )
The key concepts are considered here as the main knowledge elements that underlie the learning design and may be expected to be constructed by the pre-service mathematics teachers during the course. A knowledge element in this study is identified with the code E x .
Knowledge elements were determined based on key concepts. It is a didactical decision made by the researcher for designing the Parallel Coordinates unit.
In line with the aims of this study and the use of AiC framework, data analysis in this study are done in micro-level. In the microanalysis process the researcher did frame-by-frame coding stage to select appropriate data, continued b data categorization as the result of frame- by-frame coding stage to do focused coding. In this stage, RBC + C model is used to be the tools for analyzing the data using RBC + C table. Finally, the researcher created more general categories; found similarities and dissimilarities as well as patterns among categories; and analyzed the data quantitatively if needed in analytic coding stage. After selecting the frames, the researcher formulated four episodes in order to present the answer to the first question in this study. In every episode, RBC + C method was used to analyze the abstraction process that took place in group context and compared it with the classroom discussion. The researcher provided at most two result analysis from two different groups in every episode.
In order to answer the second research question, the researcher did analysis using data from test result in the topic of Parallel Coordinates and constant comparative technique, which was followed by data microanalysis of interview transcriptions.
One of the criteria for trustworthiness adopted in this study is credibility. In this study, specific procedures that are AiC designed and RBC + C model has been selected from Dreyfus, Heskowitz, & Swartz (2015) used throughout the data collection, analysis, and interpretation in this study. The interview questions also designed based on Zaskis and Hazzan (1999). General qualitative approach and constant comparative analysis were used to ensure credibility as well.
Another strategy used to ensure credibility is triangulation; the researcher used video recorded transcripts, interview transcripts, participants’ documents such as their worksheet, test as well as observer field notes to triangulate and verify the findings of this study.
Peer debriefing strategy (Creswell, 2009) is another strategy used in this study to ensure credibility. The researcher met with her advisers to consult about emerging design and themes. The researcher also discussed one case thoroughly with peer researchers, who also helped verify the inter-rater reliability of the initial code list.
In order to ensure the consistency of researcher’s approach in this study, the researcher documented all processes in detail, and then shared with advisers to help evaluate the processes to confirm the consistency. The researcher also checked the result from three professional video transcribers to make sure that they do not contain obvious mistake during the transcriptions. Another procedure that used by the researcher is constantly comparing data with the codes and their definitions during the process of coding.
G. Results and Discussions
Referred to the research question, there are three main results in this study. The first is the result of mathematical abstraction of pre-service mathematics teachers, the second is the levels of mathematical abstraction of pre-service mathematics teachers, and the third is the relationship between scores on Cartesian coordinate test and mathematical abstraction test on the topic of Parallel Coordinates.
G.1 Mathematical Abstraction of Pre-Service Mathematics Teachers in Learning 2D Parallel Coordinates
There are seven knowledge elements in the topic of Parallel Coordinates need to be constructed by all Participants, E A1 ,E A2 ,E B ,E C1 ,E C2 ,E D 1 , and E D2 . The abstraction process of pre-service mathematics teachers in learning concept of Parallel Coordinates is taken place in group and whole class contexts. The construction of E A1 , E A2 , E B are described in group context. There are four cases of microanalysis in group context presented from three different There are seven knowledge elements in the topic of Parallel Coordinates need to be constructed by all Participants, E A1 ,E A2 ,E B ,E C1 ,E C2 ,E D 1 , and E D2 . The abstraction process of pre-service mathematics teachers in learning concept of Parallel Coordinates is taken place in group and whole class contexts. The construction of E A1 , E A2 , E B are described in group context. There are four cases of microanalysis in group context presented from three different
The construction of E C1 ,E C2 ,E D 1 , and E D2 are presented in whole class context due to the processes are facilitated on the module as instructional scaffolding, assistance that essentially related to the preparation and organization of the activity. In addition, spontaneously pedagogical scaffolding also found mostly during the building-with actions both in group and whole class contexts. The more complex compound of the knowledge elements, the more scaffolding needed to help participants construct the knowledge elements so that the longer time needed to accomplish the epistemic actions.
The result will be described from episode I until episode IV. In every episode, there is an illustration of a case to provide clear insight for the reader. Due to the space limitation in this summary book, not all cases are presented here.
G.1.1 Episode I: Constructing E A1 and E A2
According to the analysis of students’ answer sheets and video recording, the hypothesis that participants proposed can be classified into several types, as can be seen in Table 2. :
Table 2. The Participants’ Hypothesis in Constructing E A2
No Participants’ Hypothesis
Example of Participants’
The Number
Answer
of Students
1 Participants drew a line segment 15
as the representation of a point
A (3,2) in 2D Parallel Coordinates and in the explanation they explicitly said the line segment was derived from
having
correspondence between the axes in Cartesian coordinate and axes
in Parallel Coordinates
4 of a point A(3,2) in 2D Parallel
2 Participants drew a representation x
Coordinates as an intersection of 3 two segments. In the explanation, 2
she had two hypotheses. She predicted the representation is a point or a line
3 Participants drew the
14 representation of a point A(3,2) in
2D Parallel Coordinates as a line. They explained that the line was
resulted from connecting a point- 2
3 in and a point- 2 in .
1 hypothesis that the representation of point A(3,2) in 2D Parallel
4 A participant proposed her
Coordinates is a trapezoid, so it is 2
a plane.
No Participants’ Hypothesis
Example of Participants’
The Number
Answer
of Students
5 Participants drew the
12 representation of A(3,2) in 2D Parallel Coordinates as a line
segment but in the explanation
they mentioned that the representation was a line
Total
46 *The number of participants are 45, but there is a participant proposed two hypotheses.
G.1.2 The Result of Microanalysis from Group 2
The data that used in microanalysis of Group 2 were taken from the Video Transcript 1 In Task 1, the participants were asked to predict representation of an ordered pair of Real number (3,2) in Parallel Coordinates. In this case, all the participants already have a construct in Cartesian coordinate that the representation of an ordered pair of Real number is a point. Group 2 consists of 4 participants, two participants are from high category, and two others are from mediocre category based on the result of prior knowledge test.
Table 3. Table of RBC Analysis-1 of Group 2
Lines Subject
Transcription
Epistemic Actions R
3 S[3]
I think x one [she meant that -axes ] is related to x and x two D 1 [she meant that -axes ] is related to y
4 S[23] That is equivalent D 1
5 S[3] Yup that’s right
E A1
6 L Write your name in individual worksheet [...] then
predict if we have a point, three points two in Cartesian coordinate,
6 L what is the representation of its point in Parallel Coordinates? As I explained before that there are correspondence between the xy-Axes in Cartesian coordinate and x-one, x -two Axes in Parallel Coordinates
7 S[8] This coordinate is ...
8 S[3] Hmm...[looking at the problem]
9 S[23]
I think it should be straight, like this [put her hands D 2 D 2 parallel in front of her face ]
10 S[8] One, two, negative one, ... [looking at his worksheet]
11 S[23] But it is like this
12 S[3] Maybe it’s a line [Just guessing]
13 S[23] Do you think it’s a skew line?...
14 S[8] Look at this, it is also a point [cut off the discussion between S3 and S23 and showed his result ]
Lines Subject
Transcription
Epistemic Actions R
Silent
15 L Please do the task one first [talking to the classroom]
16 S23 Yes, I think the point also like that [starring the answer D 4 of S3]... But I guess that is a plane
17 S3 How come it can be a plane?
D 4 Silent
As can be seen in Table 3, there were two participants in Group 2 who recognized the concept of “relation” between the axes as the components of Cartesian coordinate and Parallel Coordinates in order to construct E A 1 . It can be seen when they consolidated E A 1 in order to construct E A 2 and E B . The abstraction processes in constructing E A 1 that took place in Group 2 started by recognizing the components of the 2D Cartesian coordinate then building- with by took possibility of a correspondence between x-axis and ̅ - axis, y-axis and ̅ (xy- axes in Cartesian coordinate and ̅ ̅ -axes).
Referred to Table 3. lines 7 to 17, it can be inferred that S3 and S23 had recognized and built-with notions that the representation of an ordered pair of integer number (3,2) was not a point like in Cartesian coordinate, but it could be a line or plane in 2D Parallel Coordinates. Unfortunately, they are still struggling in finding the argumentation for justifying their thinking.
Table 4. Table of RBC Analysis-2 of Group 2
Lines Subject
Transcription
Epistemic Actions R
23 S[8] Look at this axis, if this axis is mapped parallel, this D 1 axis will be drawn here, isn’t it? Oh that is the
D 5 application of transformation [he meant that drawn the x-axis
24 S[3] Yup transformation? [as if shouting]
25 SS He..he..he... [all the group members laugh]
26 S[3] So how?
27 S[8] Just drawn the x-axis here
28 S[35] Hm,... I see
29 L Do not have to answer directly correct, please make
a prediction first [The lecturer remained the Students]
30 S[23] Oh,... I see if it is rotated from this angle..... [talking D 5 to herself]
31 [all the group members working on their worksheet did sketch using ruler]
32 L Could you explain how you can come to this answer? [Asking to S[8]]
33 S[8] There is a transformation from a ... Cartesian D 5 coordinate to Parallel Coordinates, in Cartesian
D 6 coordinate the axes are perpendicular while in Parallel Coordinates, the axes are parallel, my presumption is that there is a movement of the point [tried to explain his idea]
34 S3 Yes, a movement
Lines Subject
Transcription
Epistemic Actions R
35 L That’s ok, so what is a representation of the point A in 2D Parallel Coordinates?
36 S[8]
A line
E A2
37 L
A line or a line segment?
E A2
38 S[3] Segment
E A2